Weighing neutrinos using high redshift galaxy luminosity functions

Laboratory experiments measuring neutrino oscillations indicate small mass differences between different mass eigenstates of neutrinos. The absolute mass scale is however not determined, with at present the strongest upper limits coming from astronomical observations rather than terrestrial experiments. The presence of massive neutrinos suppresses the growth of perturbations below a characteristic mass scale, thereby leading to a decreased abundance of collapsed dark matter halos. Here we show that this effect can significantly alter the predicted luminosity function (LF) of high redshift galaxies. In particular we demonstrate that a stringent constraint on the neutrino mass can be obtained using the well measured galaxy LF and our semianalytic structure formation models. Combining the constraints from the Wilkinson Microwave Anisotropy Probe 7 yr (WMAP7) data with the LF data at $z\sim 4$, we get a limit on the sum of the masses of 3 degenerate neutrinos $\Sigma m_{\nu }<0.52\mathrm{eV}$ at the 95% C.L. The additional constraints using the prior on Hubble constant strengthens this limit to $\Sigma m_{\nu }<0.29\mathrm{eV}$ at the 95% C.L. This neutrino mass limit is a factor $\sim 4$ improvement compared to the constraint based on the WMAP7 data alone, and as stringent as known limits based on other astronomical observations. As different astronomical measurements may suffer from different set of biases, the method presented here provides a complementary probe of $\Sigma m_{\nu }$. We suggest that repeating this exercise on well measured luminosity functions over different redshift ranges can provide independent and tighter constraints on $\Sigma m_{\nu }$.

Figure 1

Dependence of ${\delta }_c$ on various parameters. The top panel shows ${\delta }_c$ as a function of ${\Omega }_{\Lambda }$ for a $\Lambda \mathrm{CDM}$ universe without any massive neutrinos, with $z_c=0$. The middle panel shows ${\delta }_c$ as a function of $\Sigma m_{\nu }$ for a $\Lambda \mathrm{CDM}$ universe with massive neutrinos, where $z_c=4$. The cosmological parameters are chosen according to our fiducial model (${\Omega }_{\Lambda }=0.735$, $h=0.71$) and by keeping ${\Omega }_{\mathrm{CDM}}+{\Omega }_b+{\Omega }_{\nu }=0.265$. In the bottom panel the dependence of ${\delta }_c$ on $z_c$ is plotted for $\Sigma m_{\nu }=1\mathrm{eV}$ and for the cosmological parameters as above. In all the panels the solid horizontal (solid dark) line at 1.686 corresponds to ${\delta }_c$ for a universe with only CDM in it. The red dotted line shows the variation in ${\delta }_c$ with the parameter of interest.

Figure 2

The halo abundance $N_{\mathrm{ST}}$ for $\Lambda \mathrm{CDM}$ and $\Lambda \mathrm{MDM}$ models. The top panels show $N_{\mathrm{ST}}$ as a function of halo mass for $\Sigma m_{\nu }=0$ (solid black lines), $\Sigma m_{\nu }=0.5\mathrm{eV}$ (long dashed red lines), and $\Sigma m_{\nu }=1\mathrm{eV}$ (dash-dotted blue lines). The bottom panels show the ratio of $N_{\mathrm{ST}}$ for different values of $\Sigma m_{\nu }$ to that obtained taking $\Sigma m_{\nu }=0$. The left panels are for $z=0$, while the right panels are for $z=4$.

Figure 3

UV LF of LBGs at redshifts 3, 4, 5, and 6. The solid (black) line shows the predicted best fit LF for our model with $\Sigma m_{\nu }=0$. The thick dash-dotted (blue) curve shows our model predictions with $\Sigma m_{\nu }=1\mathrm{eV}$ and using the same best fit $f_{*}/\eta$ as the $\Sigma m_{\nu }=0\mathrm{eV}$ model. This is to illustrate the suppression due to massive neutrinos. The thin dash-dotted (blue) curves are best fits for the models with $\Sigma m_{\nu }=1\mathrm{eV}$. The data points (filled and open triangles) for $z=3$ are taken from 41 and for $z=4–6$ are from 42.

Figure 4

The fits to the CMBR TT power spectrum (Cls) (left panel) and the LF at $z=4$ (right panel) using the best fitted values of cosmological parameters obtained from our MCMC analysis. Solid blue lines are obtained for cosmological parameters corresponding to $\mathrm{WMAP}7+\mathrm{HST}+\mathrm{LF}$ data. The red dashed line in the left panel gives, for comparison, Cls for best fitted cosmological parameters based on only $\mathrm{WMAP}7+\mathrm{HST}$ data.

Figure 5

Various one-dimensional and two-dimensional marginalized distributions from our analysis. The top panels correspond to our fiducial LF model with feedback parameters $M_{\mathrm{AGN}}=1.8\times {10}^{12}{M}_{\odot }$ and $V_u=95\mathrm{km}{\mathrm{s}}^{-1}$. The middle panels show plots corresponding to constraining $m_{\nu }$ directly with same feedback parameters as above. The bottom panels are for the LF model with feedback parameters $M_{\mathrm{AGN}}=1.5\times {10}^{12}{M}_{\odot }$ and $V_u=105\mathrm{km}{\mathrm{s}}^{-1}$ (model FB). In each row the left panel gives the marginalized one-dimensional distribution for ${\overline{f}}_{\nu }$ or $m_{\nu }$. The vertical dashed lines correspond to 95% confidence levels. The other two panels in each row show the regions of 68% (dark color) and 95% (light color) confidence levels for $H_{\mathit{0}}$ and ${\Omega }_{\mathrm{DM}}{h}^2$ against ${\overline{f}}_{\nu }$ or $m_{\nu }$. The various contours correspond to constraints obtained using WMAP7-only (red/dark), $\mathrm{WMAP}+\mathrm{LF}$ (blue/light), and $\mathrm{WMAP}+\mathrm{HST}+\mathrm{LF}$ (green/lightest) data.

Figure 6

The $\Sigma m_{\nu }$ vs ${\chi }^2$ curve for our fiducial cosmology with massive neutrinos. Here we varied $M_{\mathrm{AGN}}$, $V_u$, and $f_{*}/\eta$ as free parameters in our model for LF. The solid black curve gives best fit ${\chi }^2$ as a function of $m_{\nu }$ after marginalizing over all the above three parameters. The number of degrees of freedom is 9.

Figure 7

The one-dimensional marginalized distributions of $f_{*}/\eta$ from our analysis using LF at $z=4$. The solid and dotted (red) curves, respectively, give the PDF of $f_{*}/\eta$ when one uses $\mathrm{WMAP}7+\mathrm{LF}$ and $\mathrm{WMAP}7+\mathrm{HST}+\mathrm{LF}$ data with the feedback parameters as in the fiducial LF model. The blue dashed and dash-dotted curves give the PDF of $f_{*}/\eta$ when $\mathrm{WMAP}7+\mathrm{LF}$ and $\mathrm{WMAP}7+\mathrm{HST}+\mathrm{LF}$ data are used, and for the LF model with feedback parameters FB.